A014551 -id:A014551 - cp's OEIS frontend

A105579 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 3, a(2) = 4.

1, 3, 4, 1, -4, -3, 8, 17, 4, -27, -32, 25, 92, 45, -136, -223, 52, 501, 400, -599, -1396, -195, 2600, 2993, -2204, -8187, -3776, 12601, 20156, -5043, -45352, -35263, 55444, 125973, 15088, -236855, -267028, 206685, 740744, 327377, -1154108, -1808859, 499360, 4117081, 3118364, -5115795, -11352520

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: famseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Cf. A002249, A014551, A078020, A105577, A105578, A105580.

Cf. Equals (1/2) [A107920(n+4) - 2*A107920(n-1) + 3 ].

Table[(3 - ((1-I*Sqrt[7])^n + (1+I*Sqrt[7])^n)/2^n)/2 // Simplify, {n, 1, 50}] (* Jean-François Alcover, Jun 04 2017 *)

a(n+1) - a(n) = A002249(n).

a(n) = 2*a(n-1)-3*a(n-2)+2*a(n-3). G.f.: (1+x+x^2)/((1-x)*(1-x+2*x^2)). [Colin Barker, Mar 27 2012]

Corrected by T. D. Noe, Nov 07 2006

A222588 Composites of the form 2^n-1 or 2^n+1 that are non-multiples of 3.

Original entry on oeis.org

65, 511, 1025, 2047, 4097, 16385, 32767, 262145, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 4294967297, 8589934591, 17179869185, 34359738367, 68719476737, 137438953471, 274877906945

Offset: 0

Walter Nissen, Feb 25 2013

Half the numbers of the proper form are divisible by 3 and thus excluded.
For 2^n-1, n must be odd to be in this sequence.
For 2^n+1, n must be even to be in this sequence.

			31 = 2^5-1 is prime and thus not a member of the sequence.
65 = 2^6+1 has 2 proper divisors, 5 and 13, thus is a(0) in the sequence.

Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 4-7, pp 69-75.

Wikipedia, Mersenne number
Wikipedia, Fermat number
George Woltman, GIMPS

Subsequence of both A014551 and A166977.

t = 2^Range[50]; u = Union[t - 1, t + 1]; Select[u, # > 1 && Mod[#, 3] != 0 && ! PrimeQ[#] &] (* T. D. Noe, Feb 26 2013 *)

A274817 a(n) = 2a(n-1) - a(n-3) + 2a(n-4) for n>3, a(0)=1, a(1)=-1, a(2)=4, a(3)=8.

Original entry on oeis.org

1, -1, 4, 8, 19, 32, 64, 125, 256, 512, 1027, 2048, 4096, 8189, 16384, 32768, 65539, 131072, 262144, 524285, 1048576, 2097152, 4194307, 8388608, 16777216, 33554429, 67108864, 134217728, 268435459, 536870912, 1073741824, 2147483645, 4294967296, 8589934592

Offset: 0

Paul Curtz, Jul 07 2016

a(n) mod 9 = 1, 8, 4, 8, 1, 5, 1, 8, 4, 8, 1, 5, ... (repeat).
Difference table for a(n):
1,   -1,   4,  8,  19, 32, ...
-2,   5,   4, 11,  13, 32, ...
7,   -1,   7,  2,  19, 29, ...
-8,   8,  -5, 17,  10, 41, ...
16, -13,  22, -7,  31, 14, ...
-29, 35, -29, 38, -17, 65, ...
... .
The recurrence of the name is valid for every line and the main diagonal which is in A014551.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A000079, A014551, A057079.

A274817:=n->2^n - sin(n*Pi/3)*(sqrt(3) + 2*sin(2*n*Pi/3)): seq(A274817(n), n=0..40); # Wesley Ivan Hurt, Jul 07 2016

Table[2^n - Sin[n*Pi/3] (Sqrt[3] + 2 Sin[2*n*Pi/3]), {n, 0, 40}] (* Wesley Ivan Hurt, Jul 07 2016 *)
LinearRecurrence[{2, 0, -1, 2}, {1, -1, 4, 8}, 100] (* G. C. Greubel, Jul 07 2016 *)

Vec((x^3+6*x^2-3*x+1)/(-2*x^4+x^3-2*x+1) + O(x^40)) \\ Colin Barker, Jul 07 2016

G.f.: (x^3+6*x^2-3*x+1) / (-2*x^4+x^3-2*x+1). - Colin Barker, Jul 07 2016
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3. - Wesley Ivan Hurt, Jul 07 2016
a(n) = 2^n - sin(n*Pi/3)*(sqrt(3) + 2*sin(2*n*Pi/3)). - Wesley Ivan Hurt, Jul 07 2016
a(n)   = 2^n - period 6: repeat [0, 3, 0, 0, -3, 0].
a(n+1) = 2*a(n) + period 6: repeat [-3, 6, 0, 3, -6, 0].
a(n+3) = -a(n) + 9*2^n.
a(n)   = A014551(n) - A057079(n).

One term corrected and more terms added by Colin Barker, Jul 07 2016

A321483 a(n) = 7*2^n + (-1)^n.

Original entry on oeis.org

8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535

Offset: 0

Paul Curtz, Nov 11 2018

Difference table:
   8, 13, 29, 55, 113, 223, 449, ...
   5, 16, 26, 58, 110, 226, 446, 898, ...
  11, 10, 32, 52, 116, 220, 452, 892, 1796, ...
  -1, 22, 20, 64, 104, 232, 440, 904, 1784, 3592, ...
      -2, 44, 40, 128, 208, 464, 880, 1808, 3568, 7184, ...
etc.
Every diagonal is a sequence of the form k*2^m.
a(n) is divisible by
.  5 if n is a term of A004767,
. 11 if n is a term of A016885,
. 13 if n is a term of A017533.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000079, A001045, A005029, A010710, A014551, A062092, A062510, A070366, A175805, A199207, A206372.
Cf. A033999, A005009.
Subsequence of A001651, A160543, A160544.

a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)

Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018

O.g.f.: (8 + 5*x) / ((1 + x)*(1 - 2*x)). - Colin Barker, Nov 11 2018
E.g.f.: exp(-x) + 7*exp(2*x). - Stefano Spezia, Nov 12 2018
a(n) = a(n-1) + 2*a(n-2).
a(n) = 2*a(n-1) + 3*(-1)^n for n>0, a(0)=8.
a(2*k) = 7*4^k + 1, a(2*k+1) = 14*4^k - 1.
a(n) = A014551(n) + A014551(n-1) + A014551(n-2).
a(n) = 2^(n+3) - 3*A001045(n).
a(n) mod 9 = A070366(n+3).
a(n) + a(n+1) = 21*2^n.

Two terms corrected, and more terms added by Colin Barker, Nov 11 2018

A083421 a(n)=2*5^n-2^n.

Original entry on oeis.org

1, 8, 46, 242, 1234, 6218, 31186, 156122, 780994, 3905738, 19530226, 97654202, 488277154, 2441398058, 12207014866, 61035123482, 305175715714, 1525878775178, 7629394269106, 38146972131962, 190734862232674

Offset: 0

Paul Barry, Apr 29 2003

Third binomial transform of A014551(n+1)

Index entries for linear recurrences with constant coefficients, signature (7,-10)

Table[2*5^n-2^n,{n,0,20}] (* or *) LinearRecurrence[{7,-10},{1,8},30] (* Harvey P. Dale, Dec 03 2015 *)

G.f.: (1+x)/((1-2x)(1-5x));
e.g.f.: 2exp(5x)-exp(2x).
a(0)=1, a(1)=8, a(n)=7*a(n-1)-10*a(n-2). - Harvey P. Dale, Dec 03 2015

A105580 a(n+3) = a(n) - a(n+1) - a(n+2); a(0) = -5, a(1) = 6, a(2) = 0.

Original entry on oeis.org

-5, 6, 0, -11, 17, -6, -22, 45, -29, -38, 112, -103, -47, 262, -318, 9, 571, -898, 336, 1133, -2367, 1570, 1930, -5867, 5507, 2290, -13664, 16881, -927, -29618, 47426, -18735, -58309, 124470, -84896, -97883, 307249, -294262, -110870, 712381, -895773, 72522, 1535632, -2503927, 1040817, 2998742

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 2tesforseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e], 1vesforseq = A000004, ForType: 1A.

			This sequence was generated using the same floretion which generated the sequences A105577, A105578, A105579, etc.. However, in this case a force transform was applied. [Specifically, (a(n)) may be seen as the result of a tesfor-transform of the zero-sequence A000004 with respect to the floretion given in the program code.]

Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A002249, A014551, A078020, A105577, A105578, A105581.

Transpose[NestList[Join[Rest[#],ListCorrelate[ {1,-1,-1}, #]]&,{-5,6,0},50]][[1]]  (* Harvey P. Dale, Mar 14 2011 *)
CoefficientList[Series[(5-x-x^2)/(x^3-x^2-x-1),{x,0,50}],x]  (* Harvey P. Dale, Mar 14 2011 *)

G.f. (5-x-x^2)/(x^3-x^2-x-1)
a(n) = A078046(n-1) - A073145(n+3).
a(n) = -5*A057597(n+2) + A057597(n+1)+A057597(n). - R. J. Mathar, Oct 25 2022

A140252 Inverse binomial transform of A140420.

Original entry on oeis.org

0, 1, 1, 7, 7, 31, 31, 127, 127, 511, 511, 2047, 2047, 8191, 8191, 32767, 32767, 131071, 131071, 524287, 524287, 2097151, 2097151, 8388607, 8388607, 33554431, 33554431, 134217727, 134217727, 536870911, 536870911

Offset: 0

Paul Curtz, Jun 23 2008

nonn

Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 23 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1, 4, -4).

Join[{0},LinearRecurrence[{1,4,-4},{1,1,7},30]] (* Harvey P. Dale, May 28 2012 *)

a(2n+1) = a(2n+2)= A083420(n).
a(n+1)-2a(n) = (-1)^n*A014551(n), n>0.
a(n+1)-2a(n)-1 = 2*(-1)^n*A131577(n).
O.g.f.: x(1+2x^2)/((2x-1)(1+2x)(x-1)). - R. J. Mathar, Aug 02 2008
a(n) = a(n-1)+4*a(n-2)-4*a(n-3), a(0)=0, a(1)=1, a(2)=1, a(3)=7. - Harvey P. Dale, May 28 2012

Edited and extended by R. J. Mathar, Aug 02 2008

A140430 Period 6: repeat [3, 2, 4, 1, 2, 0].

Original entry on oeis.org

3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2

Offset: 0

Paul Curtz, Jun 25 2008

Associate to sequence identical to half its p-th differences.
Corresponding n-th differences table:
    3,   2,   4,   1,   2,   0,   3;
   -1,   2,  -3,   1,  -2,   3,  -1;
    3,  -5,   4,  -3,   5,  -4,   3;
   -8,   9,  -7,   8,  -9,   7,  -8;
   17, -16,  15, -17,  16, -15,  17;
  -33,  31, -32,  33, -31,  32, -33;
   64, -63,  65, -64,  63, -65,  64;
Note that the main diagonal is 3 followed by A000079(n+1).
Note also the southeast diagonal 4, 1, 5, 7, 17 is 4 followed by A014551(n+1).
Note also 3*A001045(n+1), one signed and one unsigned, in two southeast diagonals.
Starting from second line, the first column is A130750 signed.
Starting from second line, the second column is A130752 signed.
Starting from second line, the third column is A130755 signed.

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A000079 (2^n), A001045 (Jacobsthal), A014551 (Jacobsthal-Lucas).

Cf. A130750, A130752, A130755.

[2 + ((-n-2) mod 3)*(-1)^n : n in [0..100]]; // Wesley Ivan Hurt, Aug 29 2014

A140430:=n->2+((-n-2) mod 3)*(-1)^n: seq(A140430(n), n=0..100); # Wesley Ivan Hurt, Aug 29 2014

CoefficientList[Series[(3 - x + 2 x^2)/((1 - x)*(1 + x^3)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 29 2014 *)
PadRight[{},120,{3,2,4,1,2,0}] (* Harvey P. Dale, Jan 21 2023 *)

a(n)=[3,2,4,1,2,0][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

From Wesley Ivan Hurt, Aug 29 2014: (Start)
G.f.: (3-x+2*x^2)/((1-x)*(1+x^3)).
a(n) = a(n-1)-a(n-3)+a(n-4);
a(n) = 2 + ((-n-2) mod 3) * (-1)^n. (End)
a(n) = (6 + 3*cos(n*Pi) + 2*sqrt(3)*sin(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 20 2016

More terms from Wesley Ivan Hurt, Aug 29 2014

A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

Original entry on oeis.org

1, 7, 10, 25, 46, 97, 190, 385, 766, 1537, 3070, 6145, 12286, 24577, 49150, 98305, 196606, 393217, 786430, 1572865, 3145726, 6291457, 12582910, 25165825, 50331646, 100663297, 201326590, 402653185, 805306366, 1610612737, 3221225470, 6442450945, 12884901886

Offset: 0

Paul Curtz, Dec 28 2016

a(n) mod 9 = period 2: repeat [1, 7].
The last digit from 7 is of period 4: repeat [7, 0, 5, 6].
The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.
a(n) is a companion to A051049(n).
With an initial 0, A051049(n) is an autosequence of the first kind.
With an initial 2, this sequence is an autosequence of the second kind.
See the reference.
Difference table:
1,   7, 10, 25, 46,  97, ... = this sequence.
6,   3, 15, 21, 51,  93, ... = 3*A014551(n)
-3, 12,  6, 30, 42, 102, ... = -3 followed by 6*A014551(n).
The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...

			a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A003945, A005010, A010688, A010710, A014551, A051049, A096045, A199116.

seq(3*2^n-(-1)^n*(1+irem(n+1,2)),n=0..32); # Peter Luschny, Dec 29 2016

LinearRecurrence[{2,1,-2},{1,7,10},50] (* Paolo Xausa, Nov 13 2023 *)

Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 28 2016

a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.
a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.
a(n) = 2*A051049(n+1) - A051049(n).
From Colin Barker, Dec 28 2016: (Start)
a(n) = 3*2^n - 2 for n even.
a(n) = 3*2^n + 1 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.
G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647, 4294967295, 8589934592

Offset: 0

Paul Curtz, Jan 16 2017

a(n) is the first sequence on three (with its first and second differences):
1, 1, 3, 8, 17, 33, 64, 127, ...;
0, 2, 5, 9, 16, 31, 63, 128, ..., that is 0 followed by A130752;
2, 3, 4, 7, 15, 32, 65, 129, ..., that is 2 followed by A130755;
1, 1, 3, 8, 17, 33, 64, 127, ..., this sequence.
The main diagonal is 2^n.
The sum of the first three lines is 3*2^n.
Alternated sum and subtraction of a(n) and its inverse binomial transform (period 3: repeat [1, 0, 2]) gives the autosequence of the first kind b(n):
   0,  1,  1,  9, 17, 35, 63, 127, ...
   1,  0,  8,  8, 18, 28, 64, 126, ...
  -1,  8,  0, 10, 10, 36, 62, 134, ...
   9, -8, 10,  0, 26, 26, 72, 118, ... .
The main diagonal is 0's. The first two upper diagonals are A259713.
The sum of the first three lines gives 9*A001045.
a(n) mod 9 gives a periodic sequence of length 6: repeat [1, 1, 3, 8, 8, 6].
a(n) = A130750(n-1) for n > 2. - Georg Fischer, Oct 23 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A000079, A001045, A005010, A007283, A014551 (a diagonal), A057079, A062510 (a diagonal), A128834, A130750, A130752, A130755, A153234, A153237, A259713.

I:=[1,1,3]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{3, -3, 2}, {1, 1, 3}, 30] (* Jean-François Alcover, Jan 16 2017 *)

Vec((1 - 2*x + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Jan 16 2017

Binomial transform of the sequence of length 3: repeat [1, 0, 2].
a(n+3) = -a(n) + 9*2^n.
a(n) = 2^n - periodic 6: repeat [0, 1, 1, 0, -1, -1, 0].
a(n+6) = a(n) + 63*2^n.
a(n+1) = 2*a(n) - period 6: repeat [1, -1, -2, -1, 1, 2].
a(n) = 2^n - 2*sin(Pi*n/3)/sqrt(3). - Jean-François Alcover and Colin Barker, Jan 16 2017
G.f.: (1 - 2*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Colin Barker, Jan 16 2017

cp's OEIS Frontend

A105579 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 3, a(2) = 4.

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A222588 Composites of the form 2^n-1 or 2^n+1 that are non-multiples of 3.

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A274817 a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3, a(0)=1, a(1)=-1, a(2)=4, a(3)=8.

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A321483 a(n) = 7*2^n + (-1)^n.

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A083421 a(n)=2*5^n-2^n.

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A105580 a(n+3) = a(n) - a(n+1) - a(n+2); a(0) = -5, a(1) = 6, a(2) = 0.

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A140252 Inverse binomial transform of A140420.

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A140430 Period 6: repeat [3, 2, 4, 1, 2, 0].

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A274817 a(n) = 2a(n-1) - a(n-3) + 2a(n-4) for n>3, a(0)=1, a(1)=-1, a(2)=4, a(3)=8.

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.